Arc Behaviour after Current Zero and RRRV

A linearly increasing voltage at a given rate of rise (dV/dt) is used after current zero to investigate the thermal interruption capability of the nozzle configuration. The value of the rate of rise of recovery voltage (dV/dt), at which the arc will just be extinguished, is commonly known as the critical rate of rise of recovery voltage (RRRV). This will be found computationally using the five flow models. The qualitative features of the arc behaviour after current zero are similar when for different values of P0 and di/dt. Unless otherwise specified, results are for P0=21.4

atm and di/dt=13 Aμs−1.

Figure 5.17 shows typical results of post-arc current computed by the Prandtl mixing length model and the standard k-epsilon model at different values of dV/dt. Results obtained by the Chen-Kim model, the RNG model and the laminar flow model are given in Figure 5.18. The axis temperature and electrical field distributions at different instants after current zero are given in Figures 5.19, 5.20, 5.21 and 5.22 respectively for the Prandtl mixing length model, the standard k-epsilon model, the Chen-Kim model and the laminar flow model. Results obtained by the RNG model are qualitatively very similar to those for the Chen-Kim model which are, therefore, not given here.

Figure 5.17. Post-arc current computed by two flow models. P0=21.4 and di/dt=13

Aμs−1. Key of the curves are:

Prandtl mixing length model: (1) dV/dt=18 kVμs−1and (2) dV/dt=19 kVμs−1; Standard k-epsilon model: (3) dV/dt=80 kVμs−1 and (4) dV/dt=85 kVμs−1.

Figure 5.18. Post-arc current computedby three flow models. P0=21.4 and di/dt=13

Aμs−1. Key of the curves are:

Laminar flow model: (1) dV/dt=0.1 kVμs−1 and (2) dV/dt=0.13 kVμs−1; Chen-Kim k-epsilon model: (3) dV/dt=0.45 kVμs−1 and dV/dt=0.5 kVμs−1; RNG k-epsilon model: (5) dV/dt=0.3 kVμs−1 and dV/dt=0.35 kVμs−1.

For the Prandtl mixing length model, when the arc is thermally extinguished, the arc temperature decays rapidly in 0.5 μs after current zero in the region of approximately 9 mm long downstream of the exit of the flat nozzle throat, i.e. from Z=5 mm to Z=14 mm (Figure 5.19(a)). It is this critical section of the arc that takes up most of the recovery voltage, where turbulent thermal conduction is mainly responsible for the rapid cooling of the arc. The electrical field of this critical section also increases rapidly with the temperature decay (Figure 5.19(b)). The standard k-epsilon model predicts a longer critical section than that predicted by the Prandtl mixing length model, which is from Z=2.5 mm to Z=14 mm (Figures 5.20(a) and 5.20(b)). The axis temperature of this critical section also decays more rapidly, which falls below 4000 K within only 0.25 μs (Figure 5.20(a)). This is due to a higher level of turbulence predicted by the standard k-epsilon model shortly before current zero in comparison with that computed by the Prandtl mixing length model as previously discussed. The RRRV computed by the standard k-epsilon model (82.5 kVμs−1) is therefore significantly higher than that obtained by the Prandtl mixing length model (18.5 kVμs−1).

(a) (b)

(c) (d)

Figure 5.19. Variations of axis temperature and electrical field with axial position at various instants after current zero obtained by the Prandtl mixng length model. (a) Axis temperature distribution and (b) electrical field distribution for dV/dt = 18 kVμs−1

(thermal clearance); (c) Axis temperature distribution and (d) electrical field distribution for dV/dt = 19 kVμs−1 (thermal reignition).

(a) (b)

(c) (d)

Figure 5.20. Variations of axis temperature and electrical field with axial position at various instants after current zero obtained by the standard k-epsilon model. (a) Axis temperature distribution and (b) electrical field distribution for dV/dt = 80 kVμs−1 (thermal clearance); (c) Axis temperature distribution and (d) electrical field distribution for dV/dt = 85 kVμs−1 (thermal reignition).

If dV/dt exceeds RRRV, temperature in the critical region still reduces immediately after current zero but this temperature decay is soon arrested as Ohmic input is pumped into a very thin core of the critical section, after which the axis temperature starts to increase. The rapid increase in axis temperature (Figures 5.19(c) and 5.20(c)) does not result in collapse of the voltage taken up by this section (as indicated by Figures 5.19(d) and 5.20(d) that electrical field is still going up) as the temperature away from the axis is still decreasing (thus arc radius is still reducing) due to thermal inertia. When the decay of temperature away from the axis has been arrested, the rate of rise of current is extremely rapid for a given dV/dt above RRRV

(Figure 5.17). Thus, the critical section is also responsible for thermal reignition when dV/dt is above RRRV. The duration before arc reignition is approximately 0.2 μs for the Prandtl mixing length model (Figure 5.17). It is even shorter for the standard k-epsilon model (Figure 5.17).

Chen-Kim and RNG models give similar results. The critical section of the arc is from Z= 8 mm to Z=14mm (Figures 5.21(a) and 5.21(b)), which is shorter than that predicted by the Prandtl mixing length model and the standard k-epsilon model. When the arc is thermally extinguished, the rate of temperature decay predicted by these two models appears to be quite slow (Figure 5.21(a)), for which the duration of thermal recovery and/or reignition is more than 5 times that predicted by the Prandtl mixing length model and 10 times that by the standard k-epsilon model. This is due to much weaker turbulence level predicted by the Chen-Kim and the RNG model as compared with the other two models. The values of computed RRRV obtained the Chen-Kim model (0.48 kVμs−1) and the RNG model (0.33 kVμs−1) are, therefore, of two orders of magnitude lower than those computed by the Prandtl mixing length model and the standard k-epsilon model.

(a) (b)

(c) (d)

Figure 5.21. Variations of axis temperature and electrical field with axial position at various instants after current zero obtained by Chen-Kim k-epsilon model. (a) Axis temperature distribution and (b) electrical field distribution for dV/dt = 0.45 kVμs−1 (thermal clearance); (c) Axis temperature distribution and (d) electrical field distribution for dV/dt = 0.5 kVμs−1 (thermal reignition).

The laminar flow model predicts that the axis temperature for the whole arc decays during thermal recovery but the rate of temperature decay is the slowest in comparison with those predicted by the other turbulence models (Figure 5.22(a)). The electrical field increases with time due to temperature decay and the contraction of arc size as a result of radial inflow (Figure 5.22(b)). The peak of the electrical field moves from the upstream region to the downstream region of the nozzle throat (Figure 5.22(b)), which is due to strong axial convection downstream of the nozzle throat that effectively cools the arc. For the reignited case, the axis temperature for the whole arc is increased substantially in 5 μs (Figure 5.22(c)). The electrical field

increases monotonically with time (caused by arc contraction) after current zero. The maximum electrical field occurs at the nozzle throat (Z = 0) within 2 μs after current zero (figure 5.22(d)), which is expected to move to the upstream region of the nozzle throat at later times (e.g. 10 μs).

(a) (b)

(c) (d)

Figure 5.22. Variations of axis temperature and electrical field with axial position at various instants after current zero obtained by the laminar flow model. (a) Axis temperature distribution and (b) electrical field distribution for dV/dt = 0.1 kVμs−1 (thermal clearance); (c) Axis temperature distribution and (d) electrical field distribution for dV/dt = 0.13 kVμs−1 (thermal reignition).

5.3 Comparison with Experiments

The computed RRRV as a function of P0 at di/dt=13 and 25 Aμs-1 are plotted in

Figure 5.23 together with the experimental results given in [5.1] for comparison. The dependence of RRRV on P0 at a given di/dt computed by the five flow models are

listed in Table 5.8.

Figure 5.23. Comparison of measured RRRV and predicted RRRV computed by five flow models.

Table 5.8. The dependence of RRRV on P0 at a given di/dt computed by five flow

models.

di/dt=13 A/μs di/dt=25 A/μs

Predictions Experiments Predictions Experiments

Laminar flow model 6 . 0 0 P RRRV 6 . 2 0 P RRRV 58 . 0 0 P RRRV 93 . 1 0 P RRRV Prandtl mixing length model 73 . 1 0 P RRRV RRRVP01.69 Standard k-epsilon model 5 . 2 0 P RRRV RRRVP02.26 Chen-Kim k-epsilon model 2 . 1 0 P RRRV RRRVP00.93 RNG k-epsilon model 15 . 1 0 P RRRV RRRVP00.66

The RRRV predicted by the Prandtl mixing length model with optimised value of turbulence parameter shows excellent agreement with experiments at di/dt=25 Aμs-1

. The model gives similar dependence of RRRV on P0 at di/dt=13 Aμs-1 to that

for di/dt= 25 Aμs-1

while experimental results indicate a much stronger pressure dependence at lower di/dt (Table 5.8). In theory, the dependence of RRRV on P0

should not be sensitive to di/dt. If the dependence of RRRV on stagnation pressure is related to di/dt, this will result in the intersection of lines in Figure 5.23. Such intersection implies that at certain pressure range RRRV for a lower di/dt will be smaller than that for a higher di/dt. This is not physical. It is well-known that the value of RRRV has a large short to short variation. The scatter of the experimental results is not mentioned in [5.1]. Taking into account of experimental uncertainties, we feel that the predicted RRRV by the Prandtl mixing length model at 13 Aμs-1 and 21.4 atm is acceptable.

The standard k-epsilon model grossly over-predicts the values of RRRV, which also show much stronger dependence on P0 at both values of di/dt (13 and 25 Aμs−1)

in comparison with the dependence predicted by the Prandtl mixing length model. The laminar flow model gives the lowest values of computed RRRV among all the five flow models, which is also significantly lower than corresponding measurements for the range of P0 and di/dt considered in the present investigation.

For both values of di/dt (13 and 25 Aμs−1), the computed RRRV is approximately proportional to square root of P0, which is consistent with the investigation of [5.4].

The Chen-Kim model and the RNG model give similar predictions of RRRV, both of which grossly under-estimate RRRV for all cases under investigation. Compared with experiments, the RRRV computed by the Chen-Kim model and the RNG model also shows much weaker dependence on P0 at di/dt =13 and 25 Aμs−1,

which is only slightly stronger than the dependence predicted by the theory based on laminar flow as indicated in the present investigation and in [5.4].

It is noted that the experimentally measured RRRV, together with the computed RRRV obtained by the Prandtl mixing length model which gives the best prediction of measured values, shows that the RRRV is proportional to the square of stagnation

pressure. This is much stronger than the pressure dependence of arc voltage for the DC arcs investigated in Chapter 4, and the arcs under quasi-steady sate discussed earlier in this chapter, which is proportional to the square root of stagnation pressure. Up to now, there appears to be no satisfactory explanation for such pressure dependence of the RRRV. It is therefore necessary to further investigate factors affecting the dependence of the RRRV on stagnation pressure. Such investigation is, however, beyond the scope of this chapter, since the main objective of this chapter is to test the selected turbulence models in order to find a suitable turbulence model in predicting the thermal interruption capability of turbulent SF6 switching arcs.

Therefore, we present the work done for this investigation and the main findings in Appendix B for reference.